CUET PG 2026 Mathematics Question Paper with Solutions

Concept:
A real symmetric matrix satisfies: \[ A^T = A \]
This means the entries are symmetric about the main diagonal.

Step 1: Count diagonal elements

There are \( n \) diagonal elements: \[ a_{11}, a_{22}, \dots, a_{nn} \]

Step 2: Count off-diagonal elements

For \( i \neq j \), we have: \[ a_{ij} = a_{ji} \]
So each pair contributes only one independent element.

Number of such pairs: \[ \frac{n(n-1)}{2} \]

Step 3: Total independent elements
\[ n + \frac{n(n-1)}{2} = \frac{n(n+1)}{2} \]

Hence, the dimension is: \[ \frac{n(n+1)}{2} \] Quick Tip: Symmetric matrix dimension = diagonal + upper triangle
= \( n + \frac{n(n-1)}{2} = \frac{n(n+1)}{2} \)

Concept: Heine–Cantor Theorem

A very important result in real analysis states:

If a function is continuous on a closed and bounded interval \( [a,b] \),
Then it is uniformly continuous on that interval.


Step 1: Understand continuity vs uniform continuity


Continuity: For every point \(x\), we can find a \(\delta\) depending on \(x\).
Uniform continuity: A single \(\delta\) works for all \(x \in [a,b]\).


Step 2: Apply the theorem

Since the function is continuous on a closed interval \( [a,b] \), it satisfies: \[ Uniform continuity on [a,b] \]

Step 3: Why closed interval matters

Closed intervals are:

Bounded
Contain all limit points

This ensures no “escape” of function behavior at endpoints.


Conclusion: \[ The function is always uniformly continuous. \] Quick Tip: Continuous on closed interval \( \Rightarrow \) Uniformly continuous (Always true!)

Concept: Definition of \(e\)

The number \(e\) is defined as: \[ e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \]

Step 1: Recognize the standard limit

The given expression is a direct standard limit used to define \(e\).

Step 2: Intuition behind the limit

This expression arises in:

Compound interest problems
Continuous growth models


As \(n\) increases, the quantity: \[ \left(1 + \frac{1}{n}\right)^n \]
approaches a fixed number.

Step 3: Numerical idea
\[ n=1 \Rightarrow 2,\quad n=2 \Rightarrow 2.25,\quad n=10 \Rightarrow 2.593,\quad n \to \infty \Rightarrow 2.718... \]

Step 4: Final value
\[ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \approx 2.718 \]


Conclusion: \[ Limit value is e \] Quick Tip: Always remember: \[ \left(1 + \frac{1}{n}\right)^n \rightarrow e \] A standard and very important limit in calculus.

Concept: Order of elements in a cyclic group

In a cyclic group of order \( n \), the number of elements of order \( d \) (where \( d \mid n \)) is given by Euler’s totient function: \[ \phi(d) \]

Step 1: Identify the group order
\[ n = 25 = 5^2 \]

Step 2: Find valid divisors

Possible orders divide 25: \[ 1, 5, 25 \]

We need elements of order \(5\).

Step 3: Apply Euler's totient function
\[ \phi(5) = 5 - 1 = 4 \]

Step 4: Interpretation

There are exactly 4 generators of the subgroup of order 5.

Conclusion: \[ Number of elements of order 5 = 4 \] Quick Tip: In cyclic groups: Number of elements of order \(d\) = \( \phi(d) \)

Concept 1: Determinant and eigenvalues

The determinant of a matrix is equal to the product of its eigenvalues: \[ \det(A) = \lambda_1 \cdot \lambda_2 \cdot \lambda_3 \]

Step 1: Compute determinant of \(A\)

Given eigenvalues: \[ 1, 2, 3 \] \[ \det(A) = 1 \times 2 \times 3 = 6 \]

Concept 2: Determinant of powers

For any square matrix: \[ \det(A^k) = (\det A)^k \]

Step 2: Apply the formula for \(A^2\)
\[ \det(A^2) = (\det A)^2 = 6^2 = 36 \]

Step 3: Alternative understanding

Eigenvalues of \(A^2\) are: \[ 1^2, 2^2, 3^2 = 1, 4, 9 \] \[ \det(A^2) = 1 \times 4 \times 9 = 36 \]

Conclusion: \[ \det(A^2) = 36 \] Quick Tip: Two useful results: Product of eigenvalues = determinant \( \det(A^k) = (\det A)^k \)

Concept: Bolzano–Weierstrass Theorem

This is a fundamental theorem in real analysis which states:

Every bounded sequence in \( \mathbb{R}^n \) has at least one convergent subsequence.


Step 1: Understand bounded sequence

A sequence \( \{x_n\} \) is bounded if: \[ \exists M > 0 such that \|x_n\| \leq M for all n \]

Step 2: Apply the theorem

If the sequence is bounded, then it cannot “escape to infinity,” so there exists a subsequence that converges.

Step 3: Conclusion
\[ Bolzano–Weierstrass Theorem guarantees convergence of a subsequence \] Quick Tip: Bounded sequence \( \Rightarrow \) Convergent subsequence (always!)

Concept: Radius of convergence using Ratio Test

For a power series: \[ \sum a_n x^n \]
the radius of convergence \(R\) is found using: \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]

Step 1: Identify coefficients
\[ a_n = \frac{1}{n!} \]

Step 2: Apply Ratio Test
\[ \left| \frac{a_{n+1}}{a_n} \right} = \frac{1}{(n+1)!} \cdot n! = \frac{1}{n+1} \]

Step 3: Take the limit
\[ \lim_{n \to \infty} \frac{1}{n+1} = 0 \]

Step 4: Interpretation

Since the limit is 0 for all \(x\), the series converges for all real \(x\).

Conclusion: \[ R = \infty \] Quick Tip: \(\sum \frac{x^n}{n!} = e^x\)
Exponential series converges for all real numbers \(x\)

Concept: Countability of sets

A set is called:

Countable if its elements can be put in one-to-one correspondence with \( \mathbb{N} \)
Uncountable if this is not possible


Step 1: Form of rational numbers

Every rational number can be written as: \[ \frac{p}{q}, \quad p \in \mathbb{Z},\; q \in \mathbb{N} \]

Step 2: Arrangement idea

We can arrange all such fractions in a grid and traverse them diagonally (Cantor’s method), ensuring every rational number is listed.

Step 3: Conclusion

Since all rational numbers can be listed in a sequence: \[ \mathbb{Q} is countable \] Quick Tip: Even though rationals are infinite, they are still countable!

Concept: Rank–Nullity Theorem

This is a fundamental theorem in linear algebra which states: \[ rank(T) + nullity(T) = \dim(V) \]

Step 1: Understand terms


Rank: Dimension of the image (range) of \(T\)
Nullity: Dimension of the kernel (null space) of \(T\)
dim(\(V\)): Dimension of domain space


Step 2: Interpretation

The theorem splits the domain space into:

Part mapped to zero (kernel)
Part mapped to image


Step 3: Conclusion
\[ rank(T) + nullity(T) = \dim(V) \] Quick Tip: Always remember: \[ Rank + Nullity = Dimension of domain \]

Concept: Commutator Subgroup

The commutator of two elements \(a, b \in G\) is: \[ [a,b] = aba^{-1}b^{-1} \]

The commutator subgroup \(G'\) (or \([G,G]\)) is generated by all such commutators.

Step 1: Abelian group definition

A group is Abelian if: \[ ab = ba \quad \forall a,b \in G \]

Step 2: Effect on commutators

If \(ab = ba\), then: \[ [a,b] = e \]
(identity element)

Step 3: Conclusion

All commutators are identity \( \Rightarrow \) commutator subgroup is trivial: \[ G' = \{e\} \] Quick Tip: Group is Abelian \( \Leftrightarrow \) Commutator subgroup is trivial

Concept: Gaussian Integral

The integral: \[ \int_{-\infty}^{\infty} e^{-x^2} dx \]
is a standard result known as the Gaussian integral.

Step 1: Define the integral

Let: \[ I = \int_{-\infty}^{\infty} e^{-x^2} dx \]

Step 2: Square the integral
\[ I^2 = \left(\int_{-\infty}^{\infty} e^{-x^2} dx \right)\left(\int_{-\infty}^{\infty} e^{-y^2} dy \right) \]
\[ = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2)} dx\,dy \]

Step 3: Convert to polar coordinates
\[ x^2 + y^2 = r^2, \quad dx\,dy = r\,dr\,d\theta \]
\[ I^2 = \int_{0}^{2\pi} \int_{0}^{\infty} e^{-r^2} r\,dr\,d\theta \]

Step 4: Evaluate the integral
\[ \int_{0}^{\infty} e^{-r^2} r\,dr = \frac{1}{2} \]
\[ I^2 = 2\pi \cdot \frac{1}{2} = \pi \]

Step 5: Final result
\[ I = \sqrt{\pi} \] Quick Tip: \[ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \] A very important standard result in probability and analysis.

Concept: Cauchy sequence and completeness


A sequence is Cauchy if its terms get arbitrarily close to each other.
A sequence converges if it approaches a limit in the space.


Step 1: Key idea

In general metric spaces: \[ Cauchy sequence \not\Rightarrow convergent \]

Step 2: Special case

If the space is complete, then: \[ Every Cauchy sequence converges \]

Step 3: Example

In \( \mathbb{Q} \), a Cauchy sequence may converge to an irrational number, which is not in \( \mathbb{Q} \), so it does not converge in that space.

Conclusion: \[ Not every Cauchy sequence is convergent (unless space is complete) \] Quick Tip: Cauchy \( \Rightarrow \) Convergent only in complete spaces

Concept: Rank of a matrix

The rank of a matrix is the maximum number of linearly independent rows or columns.

Step 1: General rule

For an \(m \times n\) matrix: \[ rank \leq \min(m,n) \]

Step 2: Apply to given matrix

Matrix size: \[ 4 \times 3 \] \[ \min(4,3) = 3 \]

Step 3: Possible values

Rank can be any integer from \(0\) up to \(3\): \[ 0,1,2,3 \]

Step 4: Interpretation


Rank \(0\): zero matrix
Rank \(3\): full column rank


Conclusion: \[ Possible ranks = 0,1,2,3 \] Quick Tip: Rank of \(m \times n\) matrix \( \leq \min(m,n)\)

Concept: Symmetric Group

The symmetric group \( S_n \) consists of all permutations of \( n \) elements.

Step 1: Formula for order
\[ |S_n| = n! \]

Step 2: Apply for \(S_3\)
\[ |S_3| = 3! = 3 \times 2 \times 1 = 6 \]

Step 3: Interpretation

There are 6 possible ways to arrange 3 distinct elements.

Conclusion: \[ |S_3| = 6 \] Quick Tip: Order of symmetric group \(S_n = n!\)

Concept: Laplace Equation

The Laplace equation is a second-order partial differential equation widely used in physics and engineering.

Step 1: Standard form in two dimensions
\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]

Step 2: Interpretation

It describes:

Steady-state heat distribution
Electrostatic potential
Fluid flow


Step 3: Identify correct option

Option (B) matches the standard Laplace equation.

Conclusion: \[ Laplace equation in 2D is \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \] Quick Tip: Laplace equation = sum of second partial derivatives equals zero